One of the fundamental and initially controversial theories of classical physics is Boltzmann's kinetic theory of gases. Instead of tracking the individual motions of billions of atoms, it describes the evolution of the probability that a particle has a certain position and velocity. The equilibrium probability distributions have been known for more than a hundred years, but it has been very difficult to understand whether and how fast convergence to equilibrium occurs.

Villani (in collaboration with Desvillettes) obtained the first results on the convergence rate for initial data not close to equilibrium. In joint work with Mouhot, he established nonlinear Landau damping for the kinetic equations of plasma physics, settling a long-standing debate. He is one of the pioneers in applications of optimal transport theory to geometric and functional inequalities.